Fuk-Nagaev inequality in smooth Banach spaces: Optimum bounds for distributions of heavy-tailed martingales

TL;DR

This paper establishes an improved Fuk-Nagaev inequality for the maximum norms of martingales in smooth Banach spaces with finite higher moments, providing sharper bounds and applications to vector-valued functions.

Contribution

It introduces a refined inequality for martingale maxima in smooth Banach spaces, enhancing existing bounds by eliminating extraneous terms and specifying exact constants.

Findings

Derived a sharper Fuk-Nagaev inequality for Banach space martingales.

Provided a new McDiarmid-type bound for vector-valued functions.

Achieved bounds with precise constants and fewer assumptions.

Abstract

We derive a Fuk-Nagaev inequality for the maxima of norms of martingale sequences in smooth Banach spaces which allow for a finite number of higher conditional moments. The bound is obtained by combining an optimization approach for a Chernoff bound due to Rio (2017) with a classical bound for moment generating functions of smooth Banach space norms by Pinelis (1994). Our result improves comparable infinite-dimensional bounds in the literature by removing unnecessary centering terms and giving precise constants. As an application, we propose a McDiarmid-type bound for vector-valued functions which allow for a uniform bound on their conditional higher moments.